
MA232A  Euclidean and nonEuclidean Geometry
Michaelmas Term 2015
Dr. David R. Wilkins
Resources related to Gauss's General Investigations
of Curved Surfaces

Resources related to Gauss's General Investigations of Curved Surfaces
“The single most important work in the history of differential
geometry is Gauss' paper of 1827, Disquisitiones generales
circa superficies curvas.”
M. Spivak, A comprehensive introduction to
differential geometry (2nd edn, 1979), vol. 2, p. 74.
The Text of Gauss's General Investigations of Curved Surfaces
The text of Gauss's paper is available, in the translation into English
by J.C. Morehead and A.M. Hildebeitel, and also in the original
Latin, at the following URLS:—
 Carl Friedrich Gauss, General Investigations of Curved Surfaces, translated by James Cadall Morehead and Adam Miller Hildebeitel, in English, Project Gutenberg, 1902)
 Carl Friedrich Gauss, Disquisitiones generales circa superficies curvas (1828, Internet Archive, in Latin)
Notes on the Text of Gauss's General Investigations
 Gauss's General Investigations: The Differential Geometry of Curved Surfaces. [N.B., These notes
are work in progress and are thus draft, provisional, incomplete
and subject to change.]
Vector Algebra and Spherical Geometry
Section~2 to Gauss's General Investigations of Curved Surfaces
contains some results concerning spherical trigonometry. Gauss
proved these using the Cosine Rule for Spherical Trigonometry,
a result that can be obtained without much difficulty using
coordinate geometry. Gauss's results can also be obtained
as fairly direct applications of the algebra of vectors in
threedimensional Euclidean space. The following material
is available:

Notes on Vector Algebra and Spherical Trigonometry
 This begins with an account of basic vector
algebra, defining and describing scalar and
vector products in threedimensional vector
algebra, relating these products to lengths of
vectors and angles between them. Standard identities
of vector algebra are derived, including the basic
properties of the scalar triple product, the
Vector Triple Product Identity and
Lagrange's Quadruple Product Identity.
These results are then applied to obtain
basic identities of spherical trigonometry
that are used or proved in Section 2 of
Gauss's General Investigations
of Curved Surfaces
Properties of Smooth Surfaces
The following notes are based on the contemporary approach to
the theory of smooth surfaces and their tangent spaces, using
theorems of real analysis that were developed in the century
following Gauss's publication of the Disquisitiones generales
circa superficies curvas in 1828:

Notes on Smooth Surfaces
 These notes begin with a summary of the definition
and basic properties of differentiability and smoothness
for functions of several real variables. This is followed
by a discussion of smooth curvilinear coordinate systems
over open sets in threedimensional Euclidean space.
The definition of smooth surfaces is introduced, and
followed by a discussion of smooth local coordinates
on a smooth surface. The tangent space to a smooth
surface at a point of that surface is then defined
and its basic properties are discussed. The notes
then explain how differentials of smooth functions
can be viewed as linear functionals on the tangent
spaces of smooth surface. Finally some applications
of the Inverse Function Theorem of real analysis
in several variables with particular relevance
to the theory of smooth surfaces are developed.
 Curvature of Smooth Surfaces in ThreeDimensional Space
 These notes develop the theory of the Gauss Map
developed in Sections 4 onwards of Gauss's
General Investigations of Curved Surfaces.
The Hyperbolic Plane
 The Hyperbolic Plane
 These notes develop the theory of conformallyflat
Riemannian metrics on open subsets of the plane, and
applies this theory in the study of the Hyperbolic Plane.
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Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.